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The Black-Scholes option pricing model provides a closed-form pricing formula $BS(\sigma)$ for a European-exercise option with price $P$. There is no closed-form inverse for it, but because it has a closed-form vega (volatility derivative) $\nu(\sigma)$, and the derivative is nonnegative, we can use the Newton-Raphson formula with confidence. Essentially, ...

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Brenner and Subrahmanyam (1988) provided a closed form estimate of IV, you can use it as the initial estimate: $$\sigma \approx \sqrt{\cfrac{2\pi}{T}} . \cfrac{C}{S}$$

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Let $t_0, t_1, \ldots, t_n$ be observation dates, where $0=t_0 < \cdots < t_n = T$, and $\{S_t \mid t \geq 0\}$ be the equity price process without dividend payments. Then the realized variance is defined by \begin{align*} \frac{252}{n}\sum_{i=1}^n \ln^2 \frac{S_{t_i}}{S_{t_{i-1}}}. \end{align*} Note that, for sufficiently small $x$, \begin{align*} \...

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There is no "plain Black Scholes implied surface" because implied volatilities come from options market prices (calls and put). If you had a whole continuum of call prices $C : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$, $(T,K) \mapsto C(T,K)$ you would get a implied volatility function $\sigma_I : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ ...

18

Consider a more financially plausible model than Black-Scholes: one where the stock can suddenly go bankrupt due to fraud, and the volatility varies over time. Neither model is perfect, but the new one (call it SVJ) will be "less wrong". Mathematically, we no longer have the Black-Scholes SDE based on a single stochastic generator $W$ $$\frac{dS}{S} = \... 18 Taking away all frictions and incomplentess of the market, the theory says that European Call and Puts do have the same implied volatility unless there is an arbitrage opportunity by put call parity$$ C(t,K) - P(t,K) = DF_t(F_t - K)\ . If you plug the Black-Scholes formula here for the prices of the call and the put, you will see that the equality only ... 16 It is a very simple procedure and yes, Newton-Raphson is used because it converges sufficiently quickly: You need to obviously supply an option pricing model such as BS. Plug in an initial guess for implied volatility -> calculate the the option price as a function of your initial iVol guess -> apply NR -> minimize the error term until it is sufficiently ... 16 You may want to first broadly categorize volatility models before comparing between them within each class, it does not make sense to compare standard deviation models with an implied vol model. I would broadly classify as follows: Historical realized volatility: Those include standard deviation (sum of squared deviations), realized range volatility ... 16 I generally agree with @dm63's answer: A convex (concave) smile around the forward usually indicates and leptokurtic (platykurtic) implied risk-neutral probability density. Both situations can or cannot admit arbitrage. I provide you with two counterexamples to your statements. A volatility smile that is concave around the forward does not necessarily ... 16 The value of a call option does not go to infinity as the volatility goes to infinity. It tends to the discounted value of the forward F=S_0 e^{(r-q)T}, which when the dividend yield is zero, corresponds to the current value of the stock price S_0. Let me explain why. The value of a call option increases with volatility as the upside to the option is ... 15 Setting aside, that it's not pure riskless arbitrage, but rather statistical arbitrage: You can extract the profit by performing continuous delta hedging. If you constantly adjust your hedge position you gain/lose money by delta hedging. Being long option (gamma long), you sell at higher prices and buy at lower ones. Over the course of time you realize ... 14 For an option with price C, the P\&L, with respect to changes of the underlying asset price S and volatility \sigma, is given by \begin{align*} P\&L = \delta \Delta S + \frac{1}{2}\gamma (\Delta S)^2 + \nu \Delta \sigma, \end{align*} where \delta, \gamma, and \nu are respectively the delta, gamma, and vega hedge ratios. Then it is clear ... 14 Partly because it's hard to get a hold of, the Arslan et. al. paper is starting to assume mythical proportions. As said by Dimitri Vulis, the general idea of the paper is set out in (one or two of) Peter Carr's papers. For the benefit of the OP and others I will try to summarize the most salient points of the paper below and also point out the assumptions ... 13 It seems that you are thinking of the volatility as some sort of standard deviation of your stock price. It is not. In the BS model, \sigma\sqrt{T} is the standard deviation of the log-return \log(\frac{S_T}{S_0}). There is no mathematical upper bound to its standard deviation. There is also no mathematical problem with returns being negative either. ... 13 Upon close reading, this appears to be 3 (interesting) questions, not one. I'm not sure if the mods have the tools needed to split it up, so I'm just going to write down the three questions as I see them and then deal with them one by one. Note, it is simpler for me to talk about variance instead of volatility. This has no material impact on the answer. ... 13 Along with Gatheral's book, I'd recommend reading Lorenzo Bergomi's "Stochastic Volatility Modelling". The first 2 chapters are available for download on his website. That being said, let me try to give you the basic picture. Below we assume that the equity forward curve F(0,t)=\Bbb{E}_0^\Bbb{Q}[S_t] is given for all t smaller than some relevant ... 12 Note that total implied variance defined as V(T,K) = T\Sigma(T,K)^2 $$should be an increasing function of T. Otherwise you have a calendar arbitrage (sell the call with shorter expiry and buy the cheap longer one). If you interpolate linearly your implied volatility is$$ \Sigma(T,K) = w\Sigma(T_i,K) + (1-w)\Sigma(T_{i+1},K) $$with weight w = \... 12 Yes it is a better way. Just take a look to figure 3, from Buss and Vilkov (2012, RFS): 12 I think it's interesting to look at this problem graphically also. I get a different answer, depending on whether the option is ITM, ATM, or OTM. In the plot below, all options have 1-year expiry, rates are set to 0.01 and spot is 100. The ITM call has strike 80, the ATM call has strike 100 and the OTM call has strike 150. I added a linear function (y = 40* ... 11 Implied volatility has very little to do with any particular pricing model, especially not much with BS. BS is a translation tool between prices and volatility, with its own many model deficiencies. I won't get into such model assumptions because my point is an entirely different one. Even the smile/smirk is entirely unrelated to the Black-Scholes model and ... 11 My try to answer this question with some other questions: Is the BS model right? No. Is it useful: yes. Taking a traded price and the BS Model there is only one input factor that is not given by the market: the implied volatility. It is a measure to compare options across time and strike. Are there better models? yes. Those that you mention: The local vol ... 10 What they gave you is Newton's formula. If you have a function f(x) then you can find the value x_0 such that f(x_0) = 0 by this method. It uses the derivative f' which in your case is the vega. Your function is:$$ f(x) = BS(x) - M $$where BS is the theoretical price with volatility x and M is the marketprice. Then f'(x) is the ... 10 I'll outline how you can estimate the (implied) real-world density function from (observed) option prices. Having found this real-world density, you can then compute all sorts of probabilities and quantify the market's expectation of future prices. Recall firstly that (European-style) options are priced as risk-neutral expectation of the discounted payoff. ... 9 A very popular choice for mean reversion is the Ornstein–Uhlenbeck process (here in discretized form):$$L_{t+1}-L_t=\alpha(L^*-L_t)+\sigma\epsilon_t$$Here you see that the level change is governed by some parameter \alpha, the mean reversion rate (or speed), and the distance between the long run mean L^* and the actual level L_t plus some noise. A ... 9 The idea of regime switching in volatility is rooted in the observation that volatility is usually fairly consistent and "mild", and occasionally very high, say during a market crash. The concept goes further, though. Not only does the volatility level differ markedly in different regimes, but the behavior of volatility does as well (degree of mean reversion,... 9 Regime switching is another way to describe structural changes in a data series. For example, an inflation timeseries may change states from ARMA to linear as the economy moves from a period of cyclical growth to prolonged recession. A stock price may, say, be determined by and correlated to the main equity index when it has a large market capitalisation ... 9 Some Notations It's easy to get lost so let's introduce some notations and let$$ \sigma : (t, S, K, \tau) \to \sigma(K,\tau; S, t)  denote the implied volatility smile prevailing at time $t$ when the spot price is $S_t=S$ for an option with strike level $K$ and time to expiry $\tau=T-t$. From here onward, we drop the $t$ argument to keep notations ...

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From an equities perspective, there are two concepts that should not be confused in my opinion and context should make the distinction self-explicit: Forward variance swap volatility (A) Forward implied volatility smile (B) I really recommend reading Bergomi's "Stochastic Volatility Modeling" which is an excellent book for equity practitioners. The topics ...

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VG belongs in the family of variance-mean mixture models. Given a horizon $T$ the distribution of log-returns $f$ is a mixture of Gaussians $f_G$ with randomised mean and variance. The randomisation density is $g$ and its mean and variance increase with $T$. For the VG process this randomised factor is Gamma-distributed. More concretely, denote with \$f_G(x;\...

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The method described in Hallerbach (2004) always worked well for me. We derive an estimator for Black-Scholes-Merton implied volatility that, when compared to the familiar Corrado & Miller [JBaF, 1996] estimator, has substantially higher approximation accuracy and extends over a wider region of moneyness.

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